find the derivative of the function and evaluate the derivative at the given value of a.\n\n$f(x)=(\\sin…

find the derivative of the function and evaluate the derivative at the given value of a.\n\n$f(x)=(\\sin x)^{\\ln 4 x} ; a=\\frac{\\pi}{2}$\n\nfind the derivative of the function.\n\n$\\frac{d}{d x}(\\sin x)^{\\ln 4 x}=\\square$\n\n(use parentheses to clearly denote the argument of each function.)

find the derivative of the function and evaluate the derivative at the given value of a.\n\n$f(x)=(\\sin x)^{\\ln 4 x} ; a=\\frac{\\pi}{2}$\n\nfind the derivative of the function.\n\n$\\frac{d}{d x}(\\sin x)^{\\ln 4 x}=\\square$\n\n(use parentheses to clearly denote the argument of each function.)

Answer

Explanation:

Step1: Take natural logarithm on both sides

Let ( y = (\sin x)^{\ln 4x}). Then (\ln y=\ln((\sin x)^{\ln 4x})=\ln 4x\cdot\ln(\sin x))

Step2: Differentiate both sides with respect to (x)

Using the product rule ((uv)^\prime = u^\prime v+uv^\prime), where (u = \ln 4x) and (v=\ln(\sin x))

  • (u^\prime=\frac{d}{dx}(\ln 4x)=\frac{4}{4x}=\frac{1}{x})
  • (v^\prime=\frac{d}{dx}(\ln(\sin x))=\frac{\cos x}{\sin x}=\cot x)

So (\frac{1}{y}\cdot y^\prime=\frac{1}{x}\cdot\ln(\sin x)+\ln 4x\cdot\cot x)

Step3: Solve for (y^\prime)

Multiply both sides by (y = (\sin x)^{\ln 4x})

(y^\prime=(\sin x)^{\ln 4x}\left(\frac{\ln(\sin x)}{x}+\ln 4x\cdot\cot x\right))

Answer:

((\sin x)^{\ln 4x}\left(\frac{\ln(\sin x)}{x}+\ln 4x\cdot\cot x\right))